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Mirrors > Home > MPE Home > Th. List > nfald2 | Structured version Visualization version GIF version |
Description: Variation on nfald 2165 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
nfald2.1 | ⊢ Ⅎ𝑦𝜑 |
nfald2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfald2 | ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfald2.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | nfnae 2318 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
3 | 1, 2 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
4 | nfald2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfald 2165 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑦𝜓) |
6 | 5 | ex 450 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓)) |
7 | nfa1 2028 | . . 3 ⊢ Ⅎ𝑦∀𝑦𝜓 | |
8 | biidd 252 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓)) | |
9 | 8 | drnf1 2329 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥∀𝑦𝜓 ↔ Ⅎ𝑦∀𝑦𝜓)) |
10 | 7, 9 | mpbiri 248 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓) |
11 | 6, 10 | pm2.61d2 172 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∀wal 1481 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 |
This theorem is referenced by: nfexd2 2332 dvelimf 2334 nfeud2 2482 nfrald 2944 nfiotad 5854 nfixp 7927 |
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